The present invention relates generally to methods and systems in the field of digital signal processing and, more particularly, to methods and systems for reconstructing quantized signals.
Faithful reproduction is the ultimate goal of digital signal processing systems in which a representation is used to reconstruct an original signal. One example of this type of digital signal processing system is provided by digital audio systems which seek to reconstruct an analog sound signal from a digital representation. The original signal is initially digitized by, for example, sampling and quantizing to reduce the amount of information to be stored as the original signal's representation. Sampling refers to a process wherein measurements of one or more signal parameters are performed at predetermined time intervals. Quantization is the process of expressing some infinitely variable quantity by discrete or stepped values. For example, a step ladder can be said to quantize height. This is illustrated in FIG. 1, wherein the line 10 is constituted by a continuum of height values, whereas the vertical grid 20 only expresses height in jumps or quanta denoted by the hashmarks. To quantize all of the values in line 10 using the quanta defined by the grid 20, each point in the line is assigned a new height value equal to that of the nearest hashmark. Those skilled in the art will appreciate that although this example portrays the quanta as being equidistant, some applications may benefit from quanta which are of different size. Quantizing the line using the grid 20 results in the staircase function 30 shown in FIG. 2.
Clearly, the staircase function 30 and the line 10 are not the same. The difference between points in these functions is attributable to quantizing error. Quantizing error creates a bounded uncertainty region within which it is known that the original signal point lies hereinafter termed the "quantization bounds". This concept is illustrated pictorially by the shaded regions in FIG. 3.
As can be seen by comparing FIGS. 1 and 2, the quantizing error ranges from +1/2 A to -1/2 A, where A is the size of the quantum. Since the magnitude of the quantizing error A is limited, its effect can be reduced by providing more quantizing intervals, i.e., by reducing A. However, this requires that more information be provided to express each point in the original signal's representation, for example, sixteen bits instead of eight bits. Although many applications justify the additional complexity, and concomitant expense, attributable to greater quantization resolution, other applications do not. Moreover, increasing the quantization resolution merely makes the steps in the reproduced signal smaller but does not change the underlying assumption in conventional reconstruction techniques that each quantized sample represents a distinct value of the signal, rather than a range of values to which the signal is restricted during the time interval represented by the sample.
Returning to the digital audio example, one conventional method for alleviating quantizing error is to provide a low pass filter having a frequency response that prevents images from passing and allows only the baseband signal to emerge. In terms of the earlier described step function, this filter essentially smooths out the comers of the steps in the reconstructed analog signal. These low pass filters are frequently called "brickwall" filters because an ideal frequency response for such filters has a rectangular shape. However, the infinite slope at the cutoff frequency of these filters is, in practice, difficult to implement. On the other hand, more practical circuitry using low pass filters having a frequency response with a finite slope at the cutoff frequency produces unwanted aliasing products in the output signal, or filters out desirable high frequencies.
In order to overcome this problem associated with analog low pass filters, another conversion/reconstruction technique was developed known as "oversampling". Basically, an oversampled signal is one which is sampled at a rate greater than the Nyquist rate. This can be accomplished, for example, by using a digital filter to interpolate original samples to create new samples. By increasing the sampling rate, the slope of the frequency response of the low pass filter can be made more gradual. This in turn reduces the complexity, and therefore cost, of the analog reconstruction filter. Oversampling does not address the issue raised above regarding the limitations of using a staircase function to represent an original signal.
Digital filters, such as FIR filters and IIR filters, which are commonly employed in oversampling circuitry also fail to provide optimal reconstructions. These filters distribute signal energy in a fixed pattern based upon the filter's number and placement of taps, rather than adapting to local characteristics of the signal.
Thus, both of these conventional techniques suffer from the limitations inherent in the usage of a low pass filter to smooth the transition from one quantum to another. Specifically, as mentioned above, the low pass filter only serves to smooth the steps in the staircase function. This smoothed staircase, although within the uncertainty bounds imposed by quantization, is in many cases a statistically unlikely representation of the original signal. Moreover, as more smoothing is applied, the quantization bounds of the original sampled sequence are typically violated, thereby undermining the integrity of the reconstruction.
As can be seen from the foregoing, it would be desirable to provide more flexible alternatives to conventional techniques for signal reconstruction.